Since the pioneering description of the Hall coefficient of the integer quantum Hall effect (IQHE) in terms of topological invariants [1]}, the study of topological insulators has always been one of the most fascinating fields of research and innovation in modern condensed matter physics [2]}, [3]}.
On one side, topological invariants are nonlocal bulk properties of the system which can be fully classified by the symmetries of the Hamiltonian (in the absence of interaction) and allows the distinction of different gapped phases of the system.
On the other side, the presence of such topological invariants is directly related to the emergence of boundary modes (edge states).
In the IQHE, the proper topological invariant is the integral of the \(k\) -space Berry curvature over the Brillouin zone which is called Chern number and counts the number of gapless chiral edge states of the system [1]}, [5]}.
While the IQHE can occur only when time-reversal symmetry is broken, it is possible to have a topological quantum state in the presence of time-reversal symmetry which is called the quantum spin Hall effect (QSHE).
The proper topological invariant for characterizing QSHE is the \(\mathbb {Z}_2\)  invariant that is related to the number of boundary Kramer's pairs localized at the edges in a strip geometry [6]}, [7]}, [8]}.
